module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Data.Seq.Defs | {
"line": 542,
"column": 2
} | {
"line": 557,
"column": 34
} | {
"line": 559,
"column": 0
} | [
{
"pp": "α : Type u\nC : Seq α → Prop\na : α\ns : Seq α\nh1 : ∀ (b : α) (s' : Seq α), a = b ∨ C s' → C (cons b s')\nk : ℕ\ne : some a = ↑s k\n⊢ C s",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Stream'.Seq",
"Nat.recAux",
"Option.ctorIdx",
"congrA... | [] | induction k generalizing s with
| zero =>
have TH : s = cons a (tail s) := by
apply destruct_eq_cons
unfold destruct get? Functor.map
rw [← e]
rfl
rw [TH]
apply h1 _ _ (Or.inl rfl)
| succ k IH =>
cases s with
| nil => injection e
| cons b s' =>
have h_eq : (cons... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Data.Seq.Basic | {
"line": 45,
"column": 2
} | {
"line": 45,
"column": 37
} | {
"line": 47,
"column": 0
} | [
{
"pp": "α : Type u\n⊢ nil.length' = 0",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"dite_cond_eq_true",
"Stream'.Seq.terminates_nil._simp_1",
"instAddMonoidWithOneENat",
"Stream'.Seq.length'",
"ENat.instNatCast",
"Stream'.Seq.length_nil",
"instT... | [] | simp -implicitDefEqProofs [length'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Seq.Basic | {
"line": 45,
"column": 2
} | {
"line": 45,
"column": 37
} | {
"line": 47,
"column": 0
} | [
{
"pp": "α : Type u\n⊢ nil.length' = 0",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"dite_cond_eq_true",
"Stream'.Seq.terminates_nil._simp_1",
"instAddMonoidWithOneENat",
"Stream'.Seq.length'",
"ENat.instNatCast",
"Stream'.Seq.length_nil",
"instT... | [] | simp -implicitDefEqProofs [length'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Seq.Basic | {
"line": 45,
"column": 2
} | {
"line": 45,
"column": 37
} | {
"line": 47,
"column": 0
} | [
{
"pp": "α : Type u\n⊢ nil.length' = 0",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"dite_cond_eq_true",
"Stream'.Seq.terminates_nil._simp_1",
"instAddMonoidWithOneENat",
"Stream'.Seq.length'",
"ENat.instNatCast",
"Stream'.Seq.length_nil",
"instT... | [] | simp -implicitDefEqProofs [length'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Seq.Basic | {
"line": 67,
"column": 2
} | {
"line": 67,
"column": 18
} | {
"line": 69,
"column": 0
} | [
{
"pp": "α : Type u\ns : Seq α\n⊢ s.length' = 0 ↔ s = nil",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Stream'.Seq",
"False",
"NeZero.one",
"instAddMonoidWithOneENat",
"Stream'.Seq.length'",
"instLinearOrderENat",
"congrArg",
"CommSemiring... | [] | cases s <;> simp | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Data.Seq.Basic | {
"line": 67,
"column": 2
} | {
"line": 67,
"column": 18
} | {
"line": 69,
"column": 0
} | [
{
"pp": "α : Type u\ns : Seq α\n⊢ s.length' = 0 ↔ s = nil",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Stream'.Seq",
"False",
"NeZero.one",
"instAddMonoidWithOneENat",
"Stream'.Seq.length'",
"instLinearOrderENat",
"congrArg",
"CommSemiring... | [] | cases s <;> simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Seq.Basic | {
"line": 67,
"column": 2
} | {
"line": 67,
"column": 18
} | {
"line": 69,
"column": 0
} | [
{
"pp": "α : Type u\ns : Seq α\n⊢ s.length' = 0 ↔ s = nil",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Stream'.Seq",
"False",
"NeZero.one",
"instAddMonoidWithOneENat",
"Stream'.Seq.length'",
"instLinearOrderENat",
"congrArg",
"CommSemiring... | [] | cases s <;> simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Seq.Basic | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 18
} | {
"line": 73,
"column": 0
} | [
{
"pp": "α : Type u\ns : Seq α\n⊢ s.length' ≠ 0 ↔ ∃ x s', s = cons x s'",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Stream'.Seq",
"False",
"NeZero.one",
"instAddMonoidWithOneENat",
"Stream'.Seq.length'",
"and_true",
"Stream'.Seq.length'_eq_zer... | [] | cases s <;> simp | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Data.Seq.Basic | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 18
} | {
"line": 73,
"column": 0
} | [
{
"pp": "α : Type u\ns : Seq α\n⊢ s.length' ≠ 0 ↔ ∃ x s', s = cons x s'",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Stream'.Seq",
"False",
"NeZero.one",
"instAddMonoidWithOneENat",
"Stream'.Seq.length'",
"and_true",
"Stream'.Seq.length'_eq_zer... | [] | cases s <;> simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Seq.Basic | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 18
} | {
"line": 73,
"column": 0
} | [
{
"pp": "α : Type u\ns : Seq α\n⊢ s.length' ≠ 0 ↔ ∃ x s', s = cons x s'",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Stream'.Seq",
"False",
"NeZero.one",
"instAddMonoidWithOneENat",
"Stream'.Seq.length'",
"and_true",
"Stream'.Seq.length'_eq_zer... | [] | cases s <;> simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Colimit.Ring | {
"line": 91,
"column": 44
} | {
"line": 91,
"column": 76
} | {
"line": 91,
"column": 76
} | [
{
"pp": "ι : Type u_1\ninst✝³ : Preorder ι\nG : ι → Type u_2\ninst✝² : (i : ι) → CommRing (G i)\nf : (i j : ι) → i ≤ j → G i → G j\ninst✝¹ : Nonempty ι\ninst✝ : IsDirectedOrder ι\nz : FreeCommRing ((i : ι) × G i)\nx' y' : FreeCommRing ((i : ι) × G i)\ni : ι\nx : G i\nhx :\n (of G f i) x =\n (Ideal.Quotient.... | [
"ι : Type u_1\ninst✝³ : Preorder ι\nG : ι → Type u_2\ninst✝² : (i : ι) → CommRing (G i)\nf : (i j : ι) → i ≤ j → G i → G j\ninst✝¹ : Nonempty ι\ninst✝ : IsDirectedOrder ι\nz : FreeCommRing ((i : ι) × G i)\nx' y' : FreeCommRing ((i : ι) × G i)\ni : ι\nx : G i\nhx :\n (of G f i) x =\n (Ideal.Quotient.mk\n ... | rw [map_mul, of_f, of_f, hx, hy] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Seq.Computation | {
"line": 665,
"column": 2
} | {
"line": 670,
"column": 31
} | {
"line": 672,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\ns : Computation α\nf : α → Computation β\na : α\nb : β\nn : ℕ\nh2 : (f a).Results b n\nthis : a ∈ s\n⊢ ∀ (s : Computation α),\n (∀ {m : ℕ}, s.Results a m → (s.bind f).Results b (n + m)) →\n ∀ {m : ℕ}, s.think.Results a m → (s.think.bind f).Results b (n + m)",
"ppTerm... | [] | · intro _ h3 _ h1
rw [think_bind]
obtain ⟨m', h⟩ := of_results_think h1
obtain ⟨h1, e⟩ := h
rw [e]
exact results_think (h3 h1) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.Seq.Computation | {
"line": 717,
"column": 24
} | {
"line": 717,
"column": 36
} | {
"line": 719,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\ns : Computation α\nf : α → Computation β\na : α\nb : β\nh1 : s ~> a\nh2 : f a ~> b\nb' : β\nbB : b' ∈ s.bind f\na' : α\na's : a' ∈ s\nba' : b' ∈ f a\n⊢ b = b'",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"h2",
"b'",
... | [] | exact h2 ba' | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Seq.Computation | {
"line": 1056,
"column": 4
} | {
"line": 1057,
"column": 97
} | {
"line": 1058,
"column": 2
} | [
{
"pp": "case pure\nα : Type u\nβ : Type v\nR : α → β → Prop\nC : Computation α → Computation β → Prop\na : α\nb : β\n⊢ LiftRelAux R C (Sum.inl a) (pure b).destruct ↔ ∃ b_1, b_1 ∈ pure b ∧ R a b_1",
"ppTerm": "?pure",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Computation.LiftRelAux... | [] | exact
⟨fun h => ⟨_, ret_mem _, h⟩, fun ⟨b', mb, h⟩ => by rw [mem_unique (ret_mem _) mb]; exact h⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Seq.Computation | {
"line": 1056,
"column": 4
} | {
"line": 1057,
"column": 97
} | {
"line": 1058,
"column": 2
} | [
{
"pp": "case pure\nα : Type u\nβ : Type v\nR : α → β → Prop\nC : Computation α → Computation β → Prop\na : α\nb : β\n⊢ LiftRelAux R C (Sum.inl a) (pure b).destruct ↔ ∃ b_1, b_1 ∈ pure b ∧ R a b_1",
"ppTerm": "?pure",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Computation.LiftRelAux... | [] | exact
⟨fun h => ⟨_, ret_mem _, h⟩, fun ⟨b', mb, h⟩ => by rw [mem_unique (ret_mem _) mb]; exact h⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Seq.Computation | {
"line": 1056,
"column": 4
} | {
"line": 1057,
"column": 97
} | {
"line": 1058,
"column": 2
} | [
{
"pp": "case pure\nα : Type u\nβ : Type v\nR : α → β → Prop\nC : Computation α → Computation β → Prop\na : α\nb : β\n⊢ LiftRelAux R C (Sum.inl a) (pure b).destruct ↔ ∃ b_1, b_1 ∈ pure b ∧ R a b_1",
"ppTerm": "?pure",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Computation.LiftRelAux... | [] | exact
⟨fun h => ⟨_, ret_mem _, h⟩, fun ⟨b', mb, h⟩ => by rw [mem_unique (ret_mem _) mb]; exact h⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Seq.Basic | {
"line": 493,
"column": 2
} | {
"line": 493,
"column": 19
} | {
"line": 493,
"column": 19
} | [
{
"pp": "α : Type u\ns : Seq α\nn : ℕ\n⊢ s.tail.drop n = s.drop (n + 1)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Stream'.Seq",
"Stream'.Seq.drop",
"congrArg",
"id",
"instOfNatNat",
"instHAdd",
"Stream'.Seq.tail",
"HAdd... | [
"α : Type u\ns : Seq α\nn : ℕ\n⊢ s.tail.drop n = s.drop (1 + n)"
] | rw [Nat.add_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Seq.Basic | {
"line": 1006,
"column": 2
} | {
"line": 1006,
"column": 18
} | {
"line": 1008,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\na✝ : α\nf : α → Seq1 β\na : β\ns : Seq β\n⊢ join ((a, s), nil) = (a, s)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Stream'.Seq",
"congrArg",
"Prod.mk",
"Stream'.Seq.join",
"Stream'.Seq.append_nil",
"Stream'.Seq.nil"... | [] | cases s <;> simp | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.ContinuedFractions.Computation.ApproximationCorollaries | {
"line": 58,
"column": 69
} | {
"line": 59,
"column": 58
} | {
"line": 61,
"column": 0
} | [
{
"pp": "K : Type u_1\nv : K\ninst✝³ : Field K\ninst✝² : LinearOrder K\ninst✝¹ : IsStrictOrderedRing K\ninst✝ : FloorRing K\nn : ℕ\n⊢ (of v).convs (n + 1) = ↑⌊v⌋ + 1 / (of (Int.fract v)⁻¹).convs n",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"GenC... | [] | by
rw [of_convs_eq_convs', convs'_succ, of_convs_eq_convs'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.CubicDiscriminant | {
"line": 331,
"column": 89
} | {
"line": 332,
"column": 59
} | {
"line": 334,
"column": 0
} | [
{
"pp": "R : Type u_1\nP : Cubic R\ninst✝ : Semiring R\nha : P.a = 0\nhb : P.b = 0\n⊢ P.toPoly.natDegree ≤ 1",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"HMul.hMul",
"congrArg",
"Cubic.of_b_eq_zero",
"RingHom",
"id",
... | [] | by
simpa only [of_b_eq_zero ha hb] using natDegree_linear_le | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.CubicDiscriminant | {
"line": 418,
"column": 27
} | {
"line": 418,
"column": 60
} | {
"line": 419,
"column": 4
} | [
{
"pp": "F : Type u_3\nK : Type u_4\nP : Cubic F\ninst✝¹ : Field F\ninst✝ : Field K\nφ : F →+* K\nha : (map φ P).a ≠ 0\n⊢ (map φ P).toPoly.Splits ↔ (map φ P).toPoly.roots.card = 3",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.roots",
"congrArg",
... | [
"F : Type u_3\nK : Type u_4\nP : Cubic F\ninst✝¹ : Field F\ninst✝ : Field K\nφ : F →+* K\nha : (map φ P).a ≠ 0\n⊢ (map φ P).toPoly.roots.card = (map φ P).toPoly.natDegree ↔ (map φ P).toPoly.roots.card = 3"
] | Polynomial.splits_iff_card_roots, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.ContinuedFractions.ConvergentsEquiv | {
"line": 346,
"column": 10
} | {
"line": 359,
"column": 44
} | {
"line": 360,
"column": 8
} | [
{
"pp": "case refl\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : LinearOrder K\ninst✝ : IsStrictOrderedRing K\ng : GenContFract K\ngp' : Pair K\nm : ℕ\nIH :\n ∀ {g : GenContFract K},\n (∀ {gp : Pair K} {m_1 : ℕ}, m_1 < m.succ → g.s.get? m_1 = some gp → 0 < gp.a ∧ 0 < gp.b) →\n g.convs m.succ = g.convs' m.s... | [] | obtain ⟨gp_succ_m, s_succ_mth_eq⟩ : ∃ gp_succ_m, g.s.get? (m + 1) = some gp_succ_m :=
Option.ne_none_iff_exists'.mp not_terminatedAt_n
obtain ⟨gp_m, mth_s_eq⟩ : ∃ gp_m, g.s.get? m = some gp_m :=
g.s.ge_stable m.le_succ s_succ_mth_eq
-- we then plug them into the recurrence
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.ContinuedFractions.ConvergentsEquiv | {
"line": 346,
"column": 10
} | {
"line": 359,
"column": 44
} | {
"line": 360,
"column": 8
} | [
{
"pp": "case refl\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : LinearOrder K\ninst✝ : IsStrictOrderedRing K\ng : GenContFract K\ngp' : Pair K\nm : ℕ\nIH :\n ∀ {g : GenContFract K},\n (∀ {gp : Pair K} {m_1 : ℕ}, m_1 < m.succ → g.s.get? m_1 = some gp → 0 < gp.a ∧ 0 < gp.b) →\n g.convs m.succ = g.convs' m.s... | [] | obtain ⟨gp_succ_m, s_succ_mth_eq⟩ : ∃ gp_succ_m, g.s.get? (m + 1) = some gp_succ_m :=
Option.ne_none_iff_exists'.mp not_terminatedAt_n
obtain ⟨gp_m, mth_s_eq⟩ : ∃ gp_m, g.s.get? m = some gp_m :=
g.s.ge_stable m.le_succ s_succ_mth_eq
-- we then plug them into the recurrence
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.ContinuedFractions.Computation.Approximations | {
"line": 373,
"column": 8
} | {
"line": 376,
"column": 83
} | {
"line": 379,
"column": 4
} | [
{
"pp": "K : Type u_1\nv : K\nn : ℕ\ninst✝³ : Field K\ninst✝² : LinearOrder K\ninst✝¹ : IsStrictOrderedRing K\ninst✝ : FloorRing K\nifp : IntFractPair K\ng : GenContFract K := of v\nconts : Pair K := g.contsAux (n + 1)\npred_conts : Pair K := g.contsAux n\nifp_fr_ne_zero : ifp.fr ≠ 0\nA : K := conts.a\nB : K :=... | [] | subst n
simp only [succ_ne_zero, false_or] at n_eq_zero_or_not_terminatedAt_pred_n
rw [add_tsub_cancel_right] at n_eq_zero_or_not_terminatedAt_pred_n
exact (SimpContFract.of v).determinant n_eq_zero_or_not_terminatedAt_pred_n | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.ContinuedFractions.Computation.Approximations | {
"line": 373,
"column": 8
} | {
"line": 376,
"column": 83
} | {
"line": 379,
"column": 4
} | [
{
"pp": "K : Type u_1\nv : K\nn : ℕ\ninst✝³ : Field K\ninst✝² : LinearOrder K\ninst✝¹ : IsStrictOrderedRing K\ninst✝ : FloorRing K\nifp : IntFractPair K\ng : GenContFract K := of v\nconts : Pair K := g.contsAux (n + 1)\npred_conts : Pair K := g.contsAux n\nifp_fr_ne_zero : ifp.fr ≠ 0\nA : K := conts.a\nB : K :=... | [] | subst n
simp only [succ_ne_zero, false_or] at n_eq_zero_or_not_terminatedAt_pred_n
rw [add_tsub_cancel_right] at n_eq_zero_or_not_terminatedAt_pred_n
exact (SimpContFract.of v).determinant n_eq_zero_or_not_terminatedAt_pred_n | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.ContinuedFractions.Computation.Approximations | {
"line": 413,
"column": 4
} | {
"line": 414,
"column": 50
} | {
"line": 415,
"column": 2
} | [
{
"pp": "K : Type u_1\nv : K\nn : ℕ\ninst✝³ : Field K\ninst✝² : LinearOrder K\ninst✝¹ : IsStrictOrderedRing K\ninst✝ : FloorRing K\nnot_terminatedAt_n : ¬(of v).TerminatedAt n\ng : GenContFract K := of v\nnextConts : Pair K := g.contsAux (n + 2)\nconts : Pair K := g.contsAux (n + 1)\nconts_eq : conts = g.contsA... | [] | simp [nextConts, contsAux_recurrence s_nth_eq pred_conts_eq conts_eq, gp_a_eq_one,
pred_conts_eq.symm, conts_eq.symm, add_comm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.ContinuedFractions.Computation.Approximations | {
"line": 462,
"column": 4
} | {
"line": 462,
"column": 96
} | {
"line": 463,
"column": 4
} | [
{
"pp": "case right\nK : Type u_1\nv : K\nn : ℕ\ninst✝³ : Field K\ninst✝² : LinearOrder K\ninst✝¹ : IsStrictOrderedRing K\ninst✝ : FloorRing K\nnot_terminatedAt_n : ¬(of v).TerminatedAt n\ng : GenContFract K := of v\nnextConts : Pair K := g.contsAux (n + 2)\nconts : Pair K := g.contsAux (n + 1)\nconts_eq : cont... | [
"case right\nK : Type u_1\nv : K\nn : ℕ\ninst✝³ : Field K\ninst✝² : LinearOrder K\ninst✝¹ : IsStrictOrderedRing K\ninst✝ : FloorRing K\nnot_terminatedAt_n : ¬(of v).TerminatedAt n\ng : GenContFract K := of v\nnextConts : Pair K := g.contsAux (n + 2)\nconts : Pair K := g.contsAux (n + 1)\nconts_eq : conts = g.contsA... | suffices (ifp_succ_n.b : K) * conts.b ≤ ifp_n.fr⁻¹ * conts.b by rwa [← ifp_succ_n_b_eq_gp_b] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.LinearAlgebra.Eigenspace.Basic | {
"line": 288,
"column": 2
} | {
"line": 288,
"column": 7
} | {
"line": 290,
"column": 0
} | [
{
"pp": "case e'_2\nR : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsNoetherian R M\nf : End R M\nμ : R\nthis :\n iSup ⇑((f.genEigenspace μ).comp WithTop.coeOrderHom.toOrderHom) =\n monotonicSequenceLimit ((f.genEigenspace μ).comp WithTop.coeOrderHom.toOrd... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.Idempotents | {
"line": 508,
"column": 4
} | {
"line": 508,
"column": 9
} | {
"line": 509,
"column": 2
} | [
{
"pp": "case refine_3\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : Ring S\nI : Type u_3\ninst✝ : Fintype I\nf : R →+* S\ne₀ : R\nhe₀ : IsIdempotentElem e₀\nhfe₀ : RingHom.ker f = Ideal.span {e₀}\ne : I → S\nhe : CompleteOrthogonalIdempotents e\ne' : I → R\nhe' : ∀ (i : I), f (e' i) = e i\nk : I →... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.Eigenspace.Basic | {
"line": 325,
"column": 2
} | {
"line": 325,
"column": 27
} | {
"line": 326,
"column": 2
} | [
{
"pp": "R : Type v\nM : Type w\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\nμ : R\nk m : ℕ∞\nhm : 0 < m\nhk : f.HasUnifEigenvalue μ k\ncontra : Function.Injective ⇑(f - μ • 1)\nx : M\nhx : ∃ l, ↑l ≤ k ∧ x ∈ ((f - μ • 1) ^ l).ker\n⊢ x ∈ ⊥",
"ppTerm": "?m.94",
"assigned... | [
"R : Type v\nM : Type w\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\nμ : R\nk m : ℕ∞\nhm : 0 < m\nhk : f.HasUnifEigenvalue μ k\ncontra : Function.Injective ⇑(f - μ • 1)\nx : M\nl : ℕ\nhx : x ∈ ((f - μ • 1) ^ l).ker\n⊢ x ∈ ⊥"
] | rcases hx with ⟨l, -, hx⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Algebra.DirectSum.LinearMap | {
"line": 73,
"column": 2
} | {
"line": 75,
"column": 28
} | {
"line": 77,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nN : ι → Submodule R M\ninst✝³ : DecidableEq ι\ninst✝² : ∀ (i : ι), Module.Finite R ↥(N i)\ninst✝¹ : ∀ (i : ι), Free R ↥(N i)\nh : IsInternal N\ninst✝ : Fintype ι\nf : M →ₗ[R] M\nhf : ∀ (i : ι), ... | [] | simp_rw [trace_eq_matrix_trace R (h.collectedBasis b),
toMatrix_directSum_collectedBasis_eq_blockDiagonal' h h b b hf, Matrix.trace_blockDiagonal',
← trace_eq_matrix_trace] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Algebra.DirectSum.LinearMap | {
"line": 80,
"column": 2
} | {
"line": 83,
"column": 67
} | {
"line": 85,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nN : ι → Submodule R M\ninst✝² : DecidableEq ι\ninst✝¹ : ∀ (i : ι), Module.Finite R ↥(N i)\ninst✝ : ∀ (i : ι), Free R ↥(N i)\nh : IsInternal N\nhN : {i | N i ≠ ⊥}.Finite\nf : M →ₗ[R] M\nhf : ∀ (i... | [] | let _ : Fintype {i // N i ≠ ⊥} := hN.fintype
let _ : Fintype {i | N i ≠ ⊥} := hN.fintype
rw [← Finset.sum_coe_sort, trace_eq_sum_trace_restrict (isInternal_ne_bot_iff.mpr h) (hf ·)]
exact Fintype.sum_equiv hN.subtypeEquivToFinset _ _ (fun i ↦ rfl) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.DirectSum.LinearMap | {
"line": 80,
"column": 2
} | {
"line": 83,
"column": 67
} | {
"line": 85,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nN : ι → Submodule R M\ninst✝² : DecidableEq ι\ninst✝¹ : ∀ (i : ι), Module.Finite R ↥(N i)\ninst✝ : ∀ (i : ι), Free R ↥(N i)\nh : IsInternal N\nhN : {i | N i ≠ ⊥}.Finite\nf : M →ₗ[R] M\nhf : ∀ (i... | [] | let _ : Fintype {i // N i ≠ ⊥} := hN.fintype
let _ : Fintype {i | N i ≠ ⊥} := hN.fintype
rw [← Finset.sum_coe_sort, trace_eq_sum_trace_restrict (isInternal_ne_bot_iff.mpr h) (hf ·)]
exact Fintype.sum_equiv hN.subtypeEquivToFinset _ _ (fun i ↦ rfl) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.DirectSum.LinearMap | {
"line": 79,
"column": 74
} | {
"line": 83,
"column": 67
} | {
"line": 85,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nN : ι → Submodule R M\ninst✝² : DecidableEq ι\ninst✝¹ : ∀ (i : ι), Module.Finite R ↥(N i)\ninst✝ : ∀ (i : ι), Free R ↥(N i)\nh : IsInternal N\nhN : {i | N i ≠ ⊥}.Finite\nf : M →ₗ[R] M\nhf : ∀ (i... | [] | by
let _ : Fintype {i // N i ≠ ⊥} := hN.fintype
let _ : Fintype {i | N i ≠ ⊥} := hN.fintype
rw [← Finset.sum_coe_sort, trace_eq_sum_trace_restrict (isInternal_ne_bot_iff.mpr h) (hf ·)]
exact Fintype.sum_equiv hN.subtypeEquivToFinset _ _ (fun i ↦ rfl) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.DirectSum.LinearMap | {
"line": 127,
"column": 2
} | {
"line": 127,
"column": 66
} | {
"line": 128,
"column": 2
} | [
{
"pp": "R : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : IsDomain R\ninst✝² : IsPrincipalIdealRing R\ninst✝¹ : Free R M\ninst✝ : Module.Finite R M\nf g : End R M\nh_comm✝ : Commute f g\nhf : ⨆ μ, f.maxGenEigenspace μ = ⊤\nhg : ∀ (μ : R), (trace R ↥(f.maxGe... | [
"R : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : IsDomain R\ninst✝² : IsPrincipalIdealRing R\ninst✝¹ : Free R M\ninst✝ : Module.Finite R M\nf g : End R M\nh_comm✝ : Commute f g\nhf : ⨆ μ, f.maxGenEigenspace μ = ⊤\nhg : ∀ (μ : R), (trace R ↥(f.maxGenEigenspace ... | have := f.isNilpotent_restrict_maxGenEigenspace_sub_algebraMap μ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.Eigenspace.Basic | {
"line": 804,
"column": 2
} | {
"line": 806,
"column": 89
} | {
"line": 808,
"column": 0
} | [
{
"pp": "R : Type v\nM : Type w\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\np : Submodule R M\nhfp : ∀ x ∈ p, f x ∈ p\nμ : R\nhμp : Disjoint (f.eigenspace μ) p\n⊢ eigenspace (LinearMap.restrict f hfp) μ = ⊥",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [... | [] | rw [eq_bot_iff]
intro x hx
simpa using hμp.le_bot ⟨eigenspace_restrict_le_eigenspace f hfp μ ⟨x, hx, rfl⟩, x.prop⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Eigenspace.Basic | {
"line": 804,
"column": 2
} | {
"line": 806,
"column": 89
} | {
"line": 808,
"column": 0
} | [
{
"pp": "R : Type v\nM : Type w\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\np : Submodule R M\nhfp : ∀ x ∈ p, f x ∈ p\nμ : R\nhμp : Disjoint (f.eigenspace μ) p\n⊢ eigenspace (LinearMap.restrict f hfp) μ = ⊥",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [... | [] | rw [eq_bot_iff]
intro x hx
simpa using hμp.le_bot ⟨eigenspace_restrict_le_eigenspace f hfp μ ⟨x, hx, rfl⟩, x.prop⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Exact.Sequence | {
"line": 50,
"column": 4
} | {
"line": 50,
"column": 9
} | {
"line": 51,
"column": 2
} | [
{
"pp": "case refine_1\nk : Type u_1\ninst✝³ : DivisionRing k\nn : ℕ\nV : Fin (n + 2) → Type u_2\ninst✝² : (i : Fin (n + 2)) → AddCommGroup (V i)\ninst✝¹ : (i : Fin (n + 2)) → Module k (V i)\ninst✝ : ∀ (i : Fin (n + 2)), FiniteDimensional k (V i)\nf : (i : Fin (n + 1)) → V i.castSucc →ₗ[k] V i.succ\nh_exact : ∀... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Exact.Sequence | {
"line": 50,
"column": 4
} | {
"line": 50,
"column": 9
} | {
"line": 51,
"column": 2
} | [
{
"pp": "case refine_1\nk : Type u_1\ninst✝³ : DivisionRing k\nn : ℕ\nV : Fin (n + 2) → Type u_2\ninst✝² : (i : Fin (n + 2)) → AddCommGroup (V i)\ninst✝¹ : (i : Fin (n + 2)) → Module k (V i)\ninst✝ : ∀ (i : Fin (n + 2)), FiniteDimensional k (V i)\nf : (i : Fin (n + 1)) → V i.castSucc →ₗ[k] V i.succ\nh_exact : ∀... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Exact.Sequence | {
"line": 50,
"column": 4
} | {
"line": 50,
"column": 9
} | {
"line": 51,
"column": 2
} | [
{
"pp": "case refine_1\nk : Type u_1\ninst✝³ : DivisionRing k\nn : ℕ\nV : Fin (n + 2) → Type u_2\ninst✝² : (i : Fin (n + 2)) → AddCommGroup (V i)\ninst✝¹ : (i : Fin (n + 2)) → Module k (V i)\ninst✝ : ∀ (i : Fin (n + 2)), FiniteDimensional k (V i)\nf : (i : Fin (n + 1)) → V i.castSucc →ₗ[k] V i.succ\nh_exact : ∀... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Exact.Sequence | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 44
} | {
"line": 123,
"column": 2
} | [
{
"pp": "k : Type u_1\ninst✝¹⁸ : DivisionRing k\nV₀ : Type u₀\ninst✝¹⁷ : AddCommGroup V₀\ninst✝¹⁶ : Module k V₀\ninst✝¹⁵ : FiniteDimensional k V₀\nV₁ : Type u₁\ninst✝¹⁴ : AddCommGroup V₁\ninst✝¹³ : Module k V₁\ninst✝¹² : FiniteDimensional k V₁\nV₂ : Type u₂\ninst✝¹¹ : AddCommGroup V₂\ninst✝¹⁰ : Module k V₂\nins... | [
"k : Type u_1\ninst✝¹⁸ : DivisionRing k\nV₀ : Type u₀\ninst✝¹⁷ : AddCommGroup V₀\ninst✝¹⁶ : Module k V₀\ninst✝¹⁵ : FiniteDimensional k V₀\nV₁ : Type u₁\ninst✝¹⁴ : AddCommGroup V₁\ninst✝¹³ : Module k V₁\ninst✝¹² : FiniteDimensional k V₁\nV₂ : Type u₂\ninst✝¹¹ : AddCommGroup V₂\ninst✝¹⁰ : Module k V₂\ninst✝⁹ : Finite... | let W₁ := ULift.{max u₀ u₁ u₂ u₃ u₄ u₅} V₁ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.GroupTheory.FreeGroup.Reduce | {
"line": 75,
"column": 4
} | {
"line": 75,
"column": 36
} | {
"line": 76,
"column": 4
} | [
{
"pp": "case cons\nα : Type u_1\nL : List (α × Bool)\ninst✝ : DecidableEq α\nhd1 : α × Bool\ntl1 : List (α × Bool)\n⊢ Red tl1 (reduce tl1) →\n Red (hd1 :: tl1)\n (List.rec [hd1] (fun head tail tail_ih ↦ if hd1.1 = head.1 ∧ hd1.2 = !head.2 then tail else hd1 :: head :: tail)\n (reduce tl1))",
... | [
"case cons\nα : Type u_1\nL : List (α × Bool)\ninst✝ : DecidableEq α\nhd1 : α × Bool\ntl1 TL : List (α × Bool)\nhtl : reduce tl1 = TL\n⊢ Red tl1 TL →\n Red (hd1 :: tl1)\n (List.rec [hd1] (fun head tail tail_ih ↦ if hd1.1 = head.1 ∧ hd1.2 = !head.2 then tail else hd1 :: head :: tail)\n TL)"
] | generalize htl : reduce tl1 = TL | Lean.Elab.Tactic.evalGeneralize | Lean.Parser.Tactic.generalize |
Mathlib.SetTheory.Cardinal.Free | {
"line": 81,
"column": 2
} | {
"line": 89,
"column": 38
} | {
"line": 90,
"column": 2
} | [
{
"pp": "case a\nα : Type u\ninst✝ : Nonempty α\n⊢ #(FreeGroup α) ≤ max #α ℵ₀",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Cardinal.mk_list_eq_max_mk_aleph0",
"False",
"Preorder.toLT",
"Lattice.toSem... | [
"case a\nα : Type u\ninst✝ : Nonempty α\n⊢ max #α ℵ₀ ≤ #(FreeGroup α)"
] | · apply (mk_le_of_injective (FreeGroup.toWord_injective (α := α))).trans_eq
simp only [mk_list_eq_max_mk_aleph0, mk_prod, lift_uzero, mk_fintype, Fintype.card_bool,
Nat.cast_ofNat, lift_ofNat]
obtain hα | hα := lt_or_ge #α ℵ₀
· simp only [hα.le, max_eq_right, max_eq_right_iff]
exact (mul_lt_alep... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.FreeMonoid.FreeSemigroup | {
"line": 94,
"column": 15
} | {
"line": 94,
"column": 31
} | {
"line": 94,
"column": 32
} | [
{
"pp": "case coe\nα : Type u_1\na : FreeSemigroup α\n⊢ (lift fun x ↦ ↑(FreeSemigroup.of x)) ((WithOne.lift toFreeMonoid) ↑a) = ↑a",
"ppTerm": "?coe",
"assigned": true,
"usedConstants": [
"MulHom",
"WithOne",
"FreeMonoid.lift",
"MonoidHom.instMonoidHomClass",
"CancelMon... | [] | induction a with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.IntegralClosure.IntegrallyClosed | {
"line": 87,
"column": 4
} | {
"line": 87,
"column": 9
} | {
"line": 88,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nA : Type u_2\nB : Type u_3\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nhf : Function.Injective ⇑f\ninj : Function.Injective ⇑(algebraMap R B)\ncl : ∀ {x : B}, IsIntegral R x ↔ ∃ y, (algebraMap R B) y = x\n⊢ ⇑f ∘... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.IntegralClosure.IntegrallyClosed | {
"line": 89,
"column": 4
} | {
"line": 89,
"column": 9
} | {
"line": 90,
"column": 2
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝⁴ : CommRing R\nA : Type u_2\nB : Type u_3\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nhf : Function.Injective ⇑f\ninj : Function.Injective ⇑(algebraMap R B)\ncl : ∀ {x : B}, IsIntegral R x ↔ ∃ y, (algebraMap R B... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.GCDMonoid.PUnit | {
"line": 28,
"column": 18
} | {
"line": 28,
"column": 33
} | {
"line": 29,
"column": 2
} | [
{
"pp": "⊢ ∀ {a b : PUnit.{?u.2 + 1}}, a ≠ 0 → b ≠ 0 → 1 = 1 * 1",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"PUnit.commRing",
"CommSemiring.toCommMonoidWithZero",
"Units",
"Ne",
"CommMonoidWithZero.toMonoidWithZero",
"instSubsingl... | [] | by subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.GCDMonoid.PUnit | {
"line": 29,
"column": 24
} | {
"line": 29,
"column": 39
} | {
"line": 30,
"column": 2
} | [
{
"pp": "⊢ ∀ (u : PUnit.{?u.2 + 1}ˣ), 1 = u⁻¹",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"PUnit.commRing",
"CommSemiring.toCommMonoidWithZero",
"Units",
"CommMonoidWithZero.toMonoidWithZero",
"instSubsingletonPUnit",
"Units.instOne",
"Unique.i... | [] | by subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.GCDMonoid.PUnit | {
"line": 30,
"column": 29
} | {
"line": 30,
"column": 44
} | {
"line": 30,
"column": 44
} | [
{
"pp": "x✝¹ x✝ : PUnit.{?u.2 + 1}\n⊢ x✝¹ = unit * unit",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"Semigroup.toMul",
"HMul.hMul",
"PUnit.commRing",
"CommSemiring.toCommMonoidWithZero",
"SemigroupWithZero.toSemigroup",
"CommMonoidWithZero.toMonoidWi... | [] | by subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.GCDMonoid.PUnit | {
"line": 31,
"column": 30
} | {
"line": 31,
"column": 45
} | {
"line": 31,
"column": 45
} | [
{
"pp": "x✝¹ x✝ : PUnit.{?u.2 + 1}\n⊢ x✝ = unit * unit",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"Semigroup.toMul",
"HMul.hMul",
"PUnit.commRing",
"CommSemiring.toCommMonoidWithZero",
"SemigroupWithZero.toSemigroup",
"CommMonoidWithZero.toMonoidWit... | [] | by subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.GCDMonoid.PUnit | {
"line": 32,
"column": 32
} | {
"line": 32,
"column": 47
} | {
"line": 32,
"column": 47
} | [
{
"pp": "x✝⁴ x✝³ x✝² : PUnit.{?u.2 + 1}\nx✝¹ : x✝⁴ ∣ x✝²\nx✝ : x✝⁴ ∣ x✝³\n⊢ unit = x✝⁴ * unit",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"Semigroup.toMul",
"HMul.hMul",
"PUnit.commRing",
"CommSemiring.toCommMonoidWithZero",
"SemigroupWithZero.toSemigroup"... | [] | by subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.GCDMonoid.PUnit | {
"line": 33,
"column": 25
} | {
"line": 33,
"column": 40
} | {
"line": 33,
"column": 40
} | [
{
"pp": "x✝¹ x✝ : PUnit.{?u.2 + 1}\n⊢ unit * unit * ↑1 = x✝¹ * x✝",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"Units.val",
"HMul.hMul",
"MulZeroClass.toMul",
"Monoid.toMulOneClass",
"PUnit.commRing",
"CommSemiring.toCommMonoidWithZero",
"Units"... | [] | by subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.GCDMonoid.PUnit | {
"line": 34,
"column": 19
} | {
"line": 34,
"column": 34
} | {
"line": 35,
"column": 2
} | [
{
"pp": "⊢ ∀ (a : PUnit.{?u.2 + 1}), unit = 0",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"PUnit.commRing",
"CommSemiring.toCommMonoidWithZero",
"CommMonoidWithZero.toMonoidWithZero",
"instSubsingletonPUnit",
"CommRing.toCommSemiring",
"MonoidWithZer... | [] | by subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.GCDMonoid.PUnit | {
"line": 35,
"column": 20
} | {
"line": 35,
"column": 35
} | {
"line": 36,
"column": 2
} | [
{
"pp": "⊢ ∀ (a : PUnit.{?u.2 + 1}), unit = 0",
"ppTerm": "?m.63",
"assigned": true,
"usedConstants": [
"PUnit.commRing",
"CommSemiring.toCommMonoidWithZero",
"CommMonoidWithZero.toMonoidWithZero",
"instSubsingletonPUnit",
"CommRing.toCommSemiring",
"MonoidWithZer... | [] | by subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.GCDMonoid.PUnit | {
"line": 36,
"column": 19
} | {
"line": 36,
"column": 34
} | {
"line": 37,
"column": 2
} | [
{
"pp": "⊢ ∀ (a b : PUnit.{?u.2 + 1}), normalize unit = unit",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"PUnit.instStrongNormalizedGCDMonoid._proof_1",
"StrongNormalizationMonoid._proof_1",
"PUnit.commRing",
"CommSemiring.toCommMonoidWithZero",
"Units",
... | [] | by subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.GCDMonoid.PUnit | {
"line": 37,
"column": 19
} | {
"line": 37,
"column": 34
} | {
"line": 39,
"column": 0
} | [
{
"pp": "⊢ ∀ (a b : PUnit.{?u.2 + 1}), normalize unit = unit",
"ppTerm": "?m.69",
"assigned": true,
"usedConstants": [
"PUnit.instStrongNormalizedGCDMonoid._proof_1",
"StrongNormalizationMonoid._proof_1",
"PUnit.commRing",
"CommSemiring.toCommMonoidWithZero",
"Units",
... | [] | by subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Polynomial.Eisenstein.Criterion | {
"line": 168,
"column": 6
} | {
"line": 168,
"column": 29
} | {
"line": 169,
"column": 6
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsDomain R\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Algebra R K\nq f : R[X]\np : ℕ\nhq_irr : Irreducible (map (algebraMap R K) q)\nhq_monic : q.Monic\nhf_prim : f.IsPrimitive\nhfd0 : 0 < f.natDegree\nhfP : (algebraMap R K) f.leadingCoeff ≠ 0\nhfmodP... | [
"case neg\nR : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsDomain R\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Algebra R K\nq f : R[X]\np : ℕ\nhq_irr : Irreducible (map (algebraMap R K) q)\nhq_monic : q.Monic\nhf_prim : f.IsPrimitive\nhfd0 : 0 < f.natDegree\nhfP : (algebraMap R K) f.leadingCoeff ≠ 0\nhfmodP : map (alge... | simp only [← mul_assoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Polynomial.Eisenstein.Criterion | {
"line": 169,
"column": 6
} | {
"line": 169,
"column": 42
} | {
"line": 170,
"column": 4
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsDomain R\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Algebra R K\nq f : R[X]\np : ℕ\nhq_irr : Irreducible (map (algebraMap R K) q)\nhq_monic : q.Monic\nhf_prim : f.IsPrimitive\nhfd0 : 0 < f.natDegree\nhfP : (algebraMap R K) f.leadingCoeff ≠ 0\nhfmodP... | [] | exact (dvd_pow_self q hn).mul_left _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Quaternion | {
"line": 1059,
"column": 33
} | {
"line": 1059,
"column": 51
} | {
"line": 1059,
"column": 52
} | [
{
"pp": "S : Type u_1\nT : Type u_2\nR : Type u_3\ninst✝ : CommRing R\nr x✝ y✝ : R\na b x y : ℍ[R]\n| x * star x * ↑(y * star y).re",
"ppTerm": "?m.78",
"assigned": true,
"usedConstants": [
"Quaternion.coe",
"NegZeroClass.toNeg",
"Semigroup.toMul",
"HMul.hMul",
"congrAr... | [
"S : Type u_1\nT : Type u_2\nR : Type u_3\ninst✝ : CommRing R\nr x✝ y✝ : R\na b x y : ℍ[R]\n| ↑(x * star x).re * ↑(y * star y).re"
] | x.mul_star_eq_coe, | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.Algebra.Group.Action.Pointwise.Finset | {
"line": 282,
"column": 47
} | {
"line": 282,
"column": 52
} | {
"line": 283,
"column": 0
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ns : Set ℕ\ninst✝ : DecidablePred fun x ↦ x ∈ s\na n : ℕ\n⊢ (∃ y ∈ s, a + y = n) ↔ a ≤ n ∧ n - a ∈ s",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Nat.le_add_right._simp_1",
"congrArg",
"and_self",
"... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Group.ForwardDiff | {
"line": 240,
"column": 12
} | {
"line": 240,
"column": 17
} | {
"line": 241,
"column": 2
} | [
{
"pp": "case zero\nR : Type u_3\ninst✝ : CommRing R\nj : ℕ\nh : j < 0\n⊢ (Δ_[1]^[0] fun r ↦ r ^ j) = 0",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"not_lt_zero._simp_1",
"False",
"Nat.instMulZeroClass",
"LinearOrderedCommMonoidWithZero.toIsBotZeroClass",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Group.ForwardDiff | {
"line": 240,
"column": 12
} | {
"line": 240,
"column": 17
} | {
"line": 241,
"column": 2
} | [
{
"pp": "case zero\nR : Type u_3\ninst✝ : CommRing R\nj : ℕ\nh : j < 0\n⊢ (Δ_[1]^[0] fun r ↦ r ^ j) = 0",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"not_lt_zero._simp_1",
"False",
"Nat.instMulZeroClass",
"LinearOrderedCommMonoidWithZero.toIsBotZeroClass",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.ForwardDiff | {
"line": 240,
"column": 12
} | {
"line": 240,
"column": 17
} | {
"line": 241,
"column": 2
} | [
{
"pp": "case zero\nR : Type u_3\ninst✝ : CommRing R\nj : ℕ\nh : j < 0\n⊢ (Δ_[1]^[0] fun r ↦ r ^ j) = 0",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"not_lt_zero._simp_1",
"False",
"Nat.instMulZeroClass",
"LinearOrderedCommMonoidWithZero.toIsBotZeroClass",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.ForwardDiff | {
"line": 255,
"column": 12
} | {
"line": 255,
"column": 17
} | {
"line": 256,
"column": 2
} | [
{
"pp": "case zero\nR : Type u_3\ninst✝ : CommRing R\n⊢ (Δ_[1]^[0] fun r ↦ r ^ 0) = ↑0!",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Monoid.toMulOneClass",
"congrArg",
"CommSemiring.toSemiring",
"Pi.addMonoidWithOne",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Group.ForwardDiff | {
"line": 255,
"column": 12
} | {
"line": 255,
"column": 17
} | {
"line": 256,
"column": 2
} | [
{
"pp": "case zero\nR : Type u_3\ninst✝ : CommRing R\n⊢ (Δ_[1]^[0] fun r ↦ r ^ 0) = ↑0!",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Monoid.toMulOneClass",
"congrArg",
"CommSemiring.toSemiring",
"Pi.addMonoidWithOne",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.ForwardDiff | {
"line": 255,
"column": 12
} | {
"line": 255,
"column": 17
} | {
"line": 256,
"column": 2
} | [
{
"pp": "case zero\nR : Type u_3\ninst✝ : CommRing R\n⊢ (Δ_[1]^[0] fun r ↦ r ^ 0) = ↑0!",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Monoid.toMulOneClass",
"congrArg",
"CommSemiring.toSemiring",
"Pi.addMonoidWithOne",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.ForwardDiff | {
"line": 261,
"column": 4
} | {
"line": 262,
"column": 22
} | {
"line": 263,
"column": 4
} | [
{
"pp": "case succ\nR : Type u_3\ninst✝ : CommRing R\nn : ℕ\nIH : (Δ_[1]^[n] fun r ↦ r ^ n) = ↑n !\nthis : (Δ_[1] fun r ↦ r ^ (n + 1)) = ∑ i ∈ range (n + 1), (n + 1).choose i • fun r ↦ r ^ i\n⊢ (Δ_[1]^[n + 1] fun r ↦ r ^ (n + 1)) = ↑(n + 1)!",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
... | [
"case succ\nR : Type u_3\ninst✝ : CommRing R\nn : ℕ\nIH : (Δ_[1]^[n] fun r ↦ r ^ n) = ↑n !\nthis : (Δ_[1] fun r ↦ r ^ (n + 1)) = ∑ i ∈ range (n + 1), (n + 1).choose i • fun r ↦ r ^ i\n⊢ ((∑ x ∈ range n, (n + 1).choose x • Δ_[1]^[n] fun r ↦ r ^ x) + (n + 1).choose n • Δ_[1]^[n] fun r ↦ r ^ n) = ↑(n + 1)!"
] | simp_rw [iterate_succ_apply, this, fwdDiff_iter_finsetSum, fwdDiff_iter_const_smul,
sum_range_succ] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Algebra.Group.Submonoid.Finsupp | {
"line": 47,
"column": 2
} | {
"line": 47,
"column": 79
} | {
"line": 49,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝ : CommMonoid M\nι : Type u_2\nf : ι → M\na : ι →₀ ℕ\n⊢ (a.prod fun x1 x2 ↦ f x1 ^ x2) ∈ closure (Set.range f)",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Set.mem_range_self",
"Finsupp.instFunLike",
"Nat.instMulZeroClass",
"Submonoid... | [] | exact prod_mem _ fun i hi ↦ pow_mem (subset_closure (Set.mem_range_self i)) _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Group.Submonoid.Saturation | {
"line": 158,
"column": 25
} | {
"line": 158,
"column": 50
} | {
"line": 158,
"column": 50
} | [
{
"pp": "M : Type u_1\ninst✝ : MulOneClass M\nf : Set (SaturatedSubmonoid M)\na✝ b✝ : M\nhx : a✝ ∈ ⋂ s ∈ f, ↑s\nhy : b✝ ∈ ⋂ s ∈ f, ↑s\n⊢ a✝ * b✝ ∈ ⋂ s ∈ f, ↑s",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"SaturatedSubmonoid",
"congrArg",
... | [
"M : Type u_1\ninst✝ : MulOneClass M\nf : Set (SaturatedSubmonoid M)\na✝ b✝ : M\nhx : ∀ i ∈ f, a✝ ∈ ↑i\nhy : ∀ i ∈ f, b✝ ∈ ↑i\n⊢ ∀ i ∈ f, a✝ * b✝ ∈ ↑i"
] | rw [Set.mem_iInter₂] at * | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Group.Submonoid.Support | {
"line": 48,
"column": 17
} | {
"line": 48,
"column": 22
} | {
"line": 50,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nM : Submonoid G\n⊢ ∀ {x : G}, x ∈ (M ⊓ M⁻¹).carrier → x⁻¹ ∈ (M ⊓ M⁻¹).carrier",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Submonoid.inv",
"DivInvOneMonoid.toInvOneClass",
"Monoid.toMulOneClass",
"congrArg",
"and_sel... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Group.Submonoid.Support | {
"line": 48,
"column": 17
} | {
"line": 48,
"column": 22
} | {
"line": 50,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nM : Submonoid G\n⊢ ∀ {x : G}, x ∈ (M ⊓ M⁻¹).carrier → x⁻¹ ∈ (M ⊓ M⁻¹).carrier",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Submonoid.inv",
"DivInvOneMonoid.toInvOneClass",
"Monoid.toMulOneClass",
"congrArg",
"and_sel... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Submonoid.Support | {
"line": 48,
"column": 17
} | {
"line": 48,
"column": 22
} | {
"line": 50,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nM : Submonoid G\n⊢ ∀ {x : G}, x ∈ (M ⊓ M⁻¹).carrier → x⁻¹ ∈ (M ⊓ M⁻¹).carrier",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Submonoid.inv",
"DivInvOneMonoid.toInvOneClass",
"Monoid.toMulOneClass",
"congrArg",
"and_sel... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Submonoid.Support | {
"line": 61,
"column": 26
} | {
"line": 61,
"column": 31
} | {
"line": 61,
"column": 31
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nx✝³ : Subgroup G\nx✝² : Submonoid G\nx✝¹ : x✝³.toSubmonoid ≤ x✝²\nx✝ : G\n⊢ x✝ ∈ x✝³ → x✝ ∈ x✝².mulSupport",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subgroup.instSubgroupClass",
"DivInvOneMonoid.toInvOneClass",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Group.Submonoid.Support | {
"line": 61,
"column": 26
} | {
"line": 61,
"column": 31
} | {
"line": 61,
"column": 31
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nx✝³ : Subgroup G\nx✝² : Submonoid G\nx✝¹ : x✝³.toSubmonoid ≤ x✝²\nx✝ : G\n⊢ x✝ ∈ x✝³ → x✝ ∈ x✝².mulSupport",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subgroup.instSubgroupClass",
"DivInvOneMonoid.toInvOneClass",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Submonoid.Support | {
"line": 61,
"column": 26
} | {
"line": 61,
"column": 31
} | {
"line": 61,
"column": 31
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nx✝³ : Subgroup G\nx✝² : Submonoid G\nx✝¹ : x✝³.toSubmonoid ≤ x✝²\nx✝ : G\n⊢ x✝ ∈ x✝³ → x✝ ∈ x✝².mulSupport",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subgroup.instSubgroupClass",
"DivInvOneMonoid.toInvOneClass",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Submonoid.Support | {
"line": 84,
"column": 11
} | {
"line": 84,
"column": 16
} | {
"line": 85,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nM : Submonoid G\n⊢ M.IsMulPointed → M.mulSupport = ⊥",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"inv_one",
"Monoid.toMulOneClass",
"Submonoid.mul... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Group.Submonoid.Support | {
"line": 84,
"column": 11
} | {
"line": 84,
"column": 16
} | {
"line": 85,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nM : Submonoid G\n⊢ M.IsMulPointed → M.mulSupport = ⊥",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"inv_one",
"Monoid.toMulOneClass",
"Submonoid.mul... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Submonoid.Support | {
"line": 84,
"column": 11
} | {
"line": 84,
"column": 16
} | {
"line": 85,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nM : Submonoid G\n⊢ M.IsMulPointed → M.mulSupport = ⊥",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"inv_one",
"Monoid.toMulOneClass",
"Submonoid.mul... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Submonoid.Support | {
"line": 87,
"column": 4
} | {
"line": 87,
"column": 9
} | {
"line": 89,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nM : Submonoid G\nx : G\nh : (x ∈ M.mulSupport) = (x ∈ ⊥)\n⊢ x ∈ M → x⁻¹ ∈ M → x = 1",
"ppTerm": "?m.80",
"assigned": true,
"usedConstants": [
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"Monoid.toMulOneClass",
"Submonoid.mulSu... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Group.Submonoid.Support | {
"line": 109,
"column": 78
} | {
"line": 109,
"column": 83
} | {
"line": 111,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nM : Submonoid G\nhM : M.IsMulSpanning\na : G\n⊢ a ∈ M ∨ a⁻¹ ∈ M",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"hM",
"a"
],
"usedGoals": []
}
] | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Group.Submonoid.Support | {
"line": 109,
"column": 78
} | {
"line": 109,
"column": 83
} | {
"line": 111,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nM : Submonoid G\nhM : M.IsMulSpanning\na : G\n⊢ a ∈ M ∨ a⁻¹ ∈ M",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"hM",
"a"
],
"usedGoals": []
}
] | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Submonoid.Support | {
"line": 109,
"column": 78
} | {
"line": 109,
"column": 83
} | {
"line": 111,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nM : Submonoid G\nhM : M.IsMulSpanning\na : G\n⊢ a ∈ M ∨ a⁻¹ ∈ M",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"hM",
"a"
],
"usedGoals": []
}
] | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Submonoid.Support | {
"line": 113,
"column": 26
} | {
"line": 113,
"column": 31
} | {
"line": 115,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nM N : Submonoid G\nhM : M.IsMulSpanning\nh : M ≤ N\n⊢ N.IsMulSpanning",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"DivInvOneMonoid.toInvOneClass",
"Monoid.toMulOneClass",
"Group.toDivisionMonoid",
"Membership.mem",
"D... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Group.Submonoid.Support | {
"line": 113,
"column": 26
} | {
"line": 113,
"column": 31
} | {
"line": 115,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nM N : Submonoid G\nhM : M.IsMulSpanning\nh : M ≤ N\n⊢ N.IsMulSpanning",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"DivInvOneMonoid.toInvOneClass",
"Monoid.toMulOneClass",
"Group.toDivisionMonoid",
"Membership.mem",
"D... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Submonoid.Support | {
"line": 113,
"column": 26
} | {
"line": 113,
"column": 31
} | {
"line": 115,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nM N : Submonoid G\nhM : M.IsMulSpanning\nh : M ≤ N\n⊢ N.IsMulSpanning",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"DivInvOneMonoid.toInvOneClass",
"Monoid.toMulOneClass",
"Group.toDivisionMonoid",
"Membership.mem",
"D... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Submonoid.Support | {
"line": 118,
"column": 52
} | {
"line": 118,
"column": 57
} | {
"line": 118,
"column": 57
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nM : Submonoid G\nhMp : M.IsMulPointed\nhMs : M.IsMulSpanning\nN : Submonoid G\nhN : N.IsMulPointed\nh : ∀ ⦃x : G⦄, x ∈ M → x ∈ N\n⊢ ∀ ⦃x : G⦄, x ∈ N → x ∈ M",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"InvOneClass.toOne",
"DivInvOneMo... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.GroupWithZero.Action.Pointwise.Finset | {
"line": 180,
"column": 4
} | {
"line": 182,
"column": 57
} | {
"line": 184,
"column": 0
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝¹ : GroupWithZero α\ninst✝ : DecidableEq α\na : α\ns : Finset α\nha : a ≠ 0\n⊢ (s <• a)⁻¹ = a⁻¹ •> s⁻¹",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"False",
"DivInvMonoid.toInv",
"instHSMul",
... | [] | ext; simp only [mem_inv', ne_eq, MulOpposite.op_eq_zero_iff, not_false_eq_true, ←
inv_smul_mem_iff₀, MulOpposite.smul_eq_mul_unop, MulOpposite.unop_inv, MulOpposite.unop_op,
inv_eq_zero, inv_inv, smul_eq_mul, mul_inv_rev, ha] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.GroupWithZero.Action.Pointwise.Finset | {
"line": 180,
"column": 4
} | {
"line": 182,
"column": 57
} | {
"line": 184,
"column": 0
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝¹ : GroupWithZero α\ninst✝ : DecidableEq α\na : α\ns : Finset α\nha : a ≠ 0\n⊢ (s <• a)⁻¹ = a⁻¹ •> s⁻¹",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"False",
"DivInvMonoid.toInv",
"instHSMul",
... | [] | ext; simp only [mem_inv', ne_eq, MulOpposite.op_eq_zero_iff, not_false_eq_true, ←
inv_smul_mem_iff₀, MulOpposite.smul_eq_mul_unop, MulOpposite.unop_inv, MulOpposite.unop_op,
inv_eq_zero, inv_inv, smul_eq_mul, mul_inv_rev, ha] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Irreducible.Indecomposable | {
"line": 36,
"column": 2
} | {
"line": 36,
"column": 7
} | {
"line": 38,
"column": 0
} | [
{
"pp": "ι : Type u_1\nM : Type u_2\ninst✝ : Monoid M\nv : ι → M\ns : Set ι\n⊢ {i | IsMulIndecomposable v s i} ⊆ s",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"HMul.hMul",
"Monoid.toMulOneClass",
"setOf",
"Membership.mem",
"MulOne.toMu... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Group.Irreducible.Indecomposable | {
"line": 36,
"column": 2
} | {
"line": 36,
"column": 7
} | {
"line": 38,
"column": 0
} | [
{
"pp": "ι : Type u_1\nM : Type u_2\ninst✝ : Monoid M\nv : ι → M\ns : Set ι\n⊢ {i | IsMulIndecomposable v s i} ⊆ s",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"HMul.hMul",
"Monoid.toMulOneClass",
"setOf",
"Membership.mem",
"MulOne.toMu... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Irreducible.Indecomposable | {
"line": 36,
"column": 2
} | {
"line": 36,
"column": 7
} | {
"line": 38,
"column": 0
} | [
{
"pp": "ι : Type u_1\nM : Type u_2\ninst✝ : Monoid M\nv : ι → M\ns : Set ι\n⊢ {i | IsMulIndecomposable v s i} ⊆ s",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"HMul.hMul",
"Monoid.toMulOneClass",
"setOf",
"Membership.mem",
"MulOne.toMu... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Irreducible.Indecomposable | {
"line": 76,
"column": 2
} | {
"line": 76,
"column": 7
} | {
"line": 78,
"column": 0
} | [
{
"pp": "ι : Type u_1\nG : Type u_3\nS : Type u_4\ninst✝⁴ : CommGroup G\ninst✝³ : LinearOrder S\ninst✝² : InvolutiveInv ι\ninst✝¹ : CommGroup S\ninst✝ : IsOrderedMonoid S\nv : ι → G\nhv_inv : ∀ (i : ι), v i⁻¹ = (v i)⁻¹\nf : G →* S\ni : ι\nhi✝ : i ∈ {i | 1 < (invMonoidHom.comp f) (v i)}\nj : ι\nhj : j ∈ {i | 1 <... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Group.Irreducible.Indecomposable | {
"line": 109,
"column": 54
} | {
"line": 109,
"column": 59
} | {
"line": 110,
"column": 2
} | [
{
"pp": "ι : Type u_1\nM : Type u_2\nS : Type u_4\ninst✝⁴ : Monoid M\ninst✝³ : LinearOrder S\ninst✝² : Finite ι\ninst✝¹ : CommMonoid S\ninst✝ : IsOrderedCancelMonoid S\nv : ι → M\nf : M →* S\nt : Set ι := {i | IsMulIndecomposable v {j | 1 < f (v j)} i}\ns : Set ι := {j | 1 < f (v j) ∧ v j ∉ closure (v '' t)}\nh... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Group.Irreducible.Indecomposable | {
"line": 109,
"column": 54
} | {
"line": 109,
"column": 59
} | {
"line": 110,
"column": 2
} | [
{
"pp": "ι : Type u_1\nM : Type u_2\nS : Type u_4\ninst✝⁴ : Monoid M\ninst✝³ : LinearOrder S\ninst✝² : Finite ι\ninst✝¹ : CommMonoid S\ninst✝ : IsOrderedCancelMonoid S\nv : ι → M\nf : M →* S\nt : Set ι := {i | IsMulIndecomposable v {j | 1 < f (v j)} i}\ns : Set ι := {j | 1 < f (v j) ∧ v j ∉ closure (v '' t)}\nh... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Irreducible.Indecomposable | {
"line": 109,
"column": 54
} | {
"line": 109,
"column": 59
} | {
"line": 110,
"column": 2
} | [
{
"pp": "ι : Type u_1\nM : Type u_2\nS : Type u_4\ninst✝⁴ : Monoid M\ninst✝³ : LinearOrder S\ninst✝² : Finite ι\ninst✝¹ : CommMonoid S\ninst✝ : IsOrderedCancelMonoid S\nv : ι → M\nf : M →* S\nt : Set ι := {i | IsMulIndecomposable v {j | 1 < f (v j)} i}\ns : Set ι := {j | 1 < f (v j) ∧ v j ∉ closure (v '' t)}\nh... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Irreducible.Indecomposable | {
"line": 110,
"column": 54
} | {
"line": 110,
"column": 59
} | {
"line": 111,
"column": 2
} | [
{
"pp": "ι : Type u_1\nM : Type u_2\nS : Type u_4\ninst✝⁴ : Monoid M\ninst✝³ : LinearOrder S\ninst✝² : Finite ι\ninst✝¹ : CommMonoid S\ninst✝ : IsOrderedCancelMonoid S\nv : ι → M\nf : M →* S\nt : Set ι := {i | IsMulIndecomposable v {j | 1 < f (v j)} i}\ns : Set ι := {j | 1 < f (v j) ∧ v j ∉ closure (v '' t)}\nh... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Group.Irreducible.Indecomposable | {
"line": 110,
"column": 54
} | {
"line": 110,
"column": 59
} | {
"line": 111,
"column": 2
} | [
{
"pp": "ι : Type u_1\nM : Type u_2\nS : Type u_4\ninst✝⁴ : Monoid M\ninst✝³ : LinearOrder S\ninst✝² : Finite ι\ninst✝¹ : CommMonoid S\ninst✝ : IsOrderedCancelMonoid S\nv : ι → M\nf : M →* S\nt : Set ι := {i | IsMulIndecomposable v {j | 1 < f (v j)} i}\ns : Set ι := {j | 1 < f (v j) ∧ v j ∉ closure (v '' t)}\nh... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Irreducible.Indecomposable | {
"line": 110,
"column": 54
} | {
"line": 110,
"column": 59
} | {
"line": 111,
"column": 2
} | [
{
"pp": "ι : Type u_1\nM : Type u_2\nS : Type u_4\ninst✝⁴ : Monoid M\ninst✝³ : LinearOrder S\ninst✝² : Finite ι\ninst✝¹ : CommMonoid S\ninst✝ : IsOrderedCancelMonoid S\nv : ι → M\nf : M →* S\nt : Set ι := {i | IsMulIndecomposable v {j | 1 < f (v j)} i}\ns : Set ι := {j | 1 < f (v j) ∧ v j ∉ closure (v '' t)}\nh... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.GroupWithZero.Range | {
"line": 141,
"column": 17
} | {
"line": 141,
"column": 22
} | {
"line": 142,
"column": 2
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹ : MonoidWithZero A\ninst✝ : GroupWithZero B\nf : A →*₀ B\n⊢ ∀ (x y : A),\n (if h : f (x * y) = 0 then 0 else ↑⟨Units.mk0 (f (x * y)) h, ⋯⟩) =\n (if h : f x = 0 then 0 else ↑⟨Units.mk0 (f x) h, ⋯⟩) * if h : f y = 0 then 0 else ↑⟨Units.mk0 (f y) h, ⋯⟩",
"ppT... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.GroupWithZero.Range | {
"line": 141,
"column": 17
} | {
"line": 141,
"column": 22
} | {
"line": 142,
"column": 2
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹ : MonoidWithZero A\ninst✝ : GroupWithZero B\nf : A →*₀ B\n⊢ ∀ (x y : A),\n (if h : f (x * y) = 0 then 0 else ↑⟨Units.mk0 (f (x * y)) h, ⋯⟩) =\n (if h : f x = 0 then 0 else ↑⟨Units.mk0 (f x) h, ⋯⟩) * if h : f y = 0 then 0 else ↑⟨Units.mk0 (f y) h, ⋯⟩",
"ppT... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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